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Continuous flows generate few homeomorphisms

Part of: Low-dimensional dynamical systems Topological dynamics Maps and general types of spaces defined by maps Smooth dynamical systems: general theory Connections with other structures, applications

Published online by Cambridge University Press:  25 August 2020

Wescley Bonomo  and
Paulo Varandas
Wescley Bonomo
Affiliation:
Universidade Federal do Espírito Santo, CEUNES, Rodovia Governador Mario Covas, Km 60, S ao Mateus29.932-900, Brazil ([email protected])
Paulo Varandas
Affiliation:
Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador40170-110, Brazil Universidade do Porto Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, Porto4169-007, Portugal ([email protected])
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Abstract

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

Keywords

Embedding problemflowable homeomorphismstopological entropyrotation sets

MSC classification

Primary: 37C10: Vector fields, flows, ordinary differential equations 54C25: Embedding 54H20: Topological dynamics
Secondary: 37B40: Topological entropy 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 971 - 983
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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